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Journal of Oceanography, Vol. 56, pp. 261 to 273, 2000 Tidal Vorticity Around a Coastal Promontory MOON -JIN PARK 1* and D ONG-PING WANG2 1 2 Department of Oceanography, Chungnam National University, Taejon, Korea Marine Sciences Research Center, State University of New York, Stony Brook, U.S.A. (Received 28 January 1999; in revised form 28 September 1999; accepted 28 September 1999) The vorticity generation around a coastal promontory is examined using a transport vorticity equation. The complete vorticity balance analysis is made for the transient and residual vorticity generations around an idealized, symmetric promontory. The topographic vorticity tendency is found to be the dominant forcing term in the production of transient tidal vorticities. This result is different from the previous works which emphasized the effects of lateral and bottom frictions. The residual vorticity balance is between advection and topographic vorticity tendency. The model results are consistent with the observations off Gay Head, Massachusetts. Also, the stability of a promontory as well as the offshore sand bank formation are studied by examining the sand transport pattern around the promontory. Strong deposition occurs off the tip of the promontory while erosion occurs along the upstream slope of the promontory suggesting that an originally symmetric promontory tends to incline cyclonically from the axis normal to the tidal stream. Such tendency is indeed found among coastal promontories. Keywords: ⋅ Vorticity generation, ⋅ vorticity balance, ⋅ stability of coastal promontory, ⋅ topographic vorticity tendency, ⋅ symmetric promontory, ⋅ asymmetric residual eddy pair, ⋅ cyclonic inclination, ⋅ sediment transport. Yanagi (1976, 1978), Oonishi (1977), Pingree (1978), Pingree and Maddock (1979), Ridderinkhof (1989), Ridderinkhof and Zimmerman (1990), Signell and Geyer (1991) used vertically-averaged two-dimensional models to study formation of the tidal residual eddies around a headland. Tee (1976), Yanagi (1976, 1978) and Oonishi (1977) suggested that the residual eddy results from advection of the vorticities generated in a coastal boundary layer. Pingree and Maddock (1979), Ridderinkhof (1989), Ridderinkhof and Zimmerman (1990) on the other hand, indicated that the frictional torque is responsible for the vorticity generation. Geyer and Signell (1990, 1991) used a ship-board acoustic Doppler current profiler to map the structures of transient and residual tidal eddies around Gay Head, a coastal promontory located at the seaward end of Vineyard Sound, Massachusetts. The Vinyard Sound is about 20 m deep, 15 km wide, and the dimensions of Gay Head is about 5 km alongshore and 4 km offshore. Signell and Geyer (1991) simulated the observed transient eddies and suggested that the transient tidal eddies are formed when the tidal flow separates from the coast at the headland tip. Signell and Geyer (1991), however, did not examine the vorticity balance, which is necessary to fully understand the generation of tidal eddies. Previously, work on the vorticity balance has focused on the vorticity associated with the depth-averaged velocity. Park and Wang 1. Introduction Tidal currents play an important role in dispersing and transporting materials in shallow coastal seas. Strong oscillatory tidal currents produce residual currents through nonlinear interaction with the bottom topography or coastline. For example, a clockwise residual circulation is often found around a sand bank (Huthnance, 1973). Sources of the tidal residual circulation can be examined by analyzing the vorticity balance. Tee (1976), Yanagi (1976) and Pingree (1978) suggested that the nonlinear advection is the major source of the residual vorticity. Zimmerman (1978, 1980, 1981) and Robinson (1981, 1983) found that the tidal residual current is best developed when the topographic length scale is comparable to the tidal excursion length. They also indicated that the column stretching and frictional torque are the major sources for instantaneous tidal vorticities. Effects of the residual current on material dispersion were studied by Zimmerman (1976, 1978, 1986), Uncles (1982) and Loder et al. (1982). They suggested that the residual current combined with the oscillatory tidal currents can give rise to a large dispersion coefficient. Strong tidal flows also can produce residual eddies around a headland or coastal promontory. Tee (1976), * Corresponding author. E-mail: [email protected] Copyright © The Oceanographic Society of Japan. 261 (1991, 1994), on the other hand, analyzed the vorticity associated with the total transport. They showed that it is advantageous to analyze the transport vorticity whose pattern can be quite different from the vorticity of the depth-averaged velocity. They suggested that for transient tidal vorticity generated over an isolated bottom topography the topographic vorticity tendency is dominant. The tidal residual eddy may have implications on the sediment transport. For example, Huthnance (1973), Zimmerman (1981), and Park and Wang (1994) suggested a correspondence between the sand bank formation and tidal residual eddies in shallow coastal seas. Pingree (1978) and Pingree and Maddock (1979) examined the formation of a sand bank (Shambles Bank off Portland Bill, England) around a coastal promontory. They suggested that the sand bank is caused by the tidal vorticities generated around the promontory and the sand bank tends to grow faster under a cyclonic vorticity than under an anticyclonic vorticity. Huthnance (1982a, 1982b) indicated that a sand bank builds up most rapidly when it is inclined cyclonically relative to the tidal stream direction. Park and Wang (1991, 1994) obtained similar results showing that a circular hollow or bump is unstable and tends to become elliptic with its principal axis oriented cyclonically to the tidal stream. In this paper we use the transport vorticity equation to study the tidal vorticities generated around a coastal promontory. We consider an idealized case representing the conditions at Gay Head, Massachusetts. We compute the transient and residual tidal vorticities, analyze the transport vorticity balance, and compare our results with the observations (Geyer and Signell, 1990, 1991) and previous model results (Signell and Geyer, 1990, 1991). We also examine the processes responsible for offshore sand bank formation, and investigate the stability and the preferential orientation of a coastal promontory. 2. Model Tidal vorticities around a coastal promontory are investigated in an idealized coastal ocean. The promontory is situated in a shallow coastal sea and has dimensions comparable to Gay Head, Vinyard Sound, Massachusetts. The Gay Head is about 5 km alongshore and 4 km offshore and slopes to the Sound in about 2 km from the tip of the headland. The tidal velocities are dominated by the M2 tide with a typical magnitude of 60 cm s–1 (Geyer and Signell, 1990). The model ocean is oriented parallel to the x-axis and closed by northern and southern boundaries (Fig. 1). It is 30 km wide, 50 km long and 20 m deep, and opens at both ends. The model promontory is 4 km wide, 4 km long, and has a shallow, submerged flat surface of 2.2 m depth (Fig. 1). The model geometry includes essential features of Gay Head, but avoids the land boundary which may introduce artificial (i.e. numerical) 262 M.-J. Park and D.-P. Wang Fig. 1. Schematic diagram of model geometry for a coastal promontory. The shaded area is the flat top of the promontory of 2.2 m depth and the outer dashed line is the perimeter of the bottom slope. A, B, and C are the locations where the transient vorticity balance is examined. vorticities. The model is based on the vertically-integrated equations of motion and the continuity equation (Pritchard, 1971); ∂U ∂uU ∂vU + + − fV ∂y ∂t ∂x ( U U2 + V2 ∂ζ = − gH −κ ∂x H2 ) 1/ 2 + Dx (1) + Dy (2 ) ∂V ∂uV ∂vV + + + fU ∂t ∂x ∂y ( V U2 + V2 ∂ζ = − gH −κ ∂y H2 ) 1/ 2 ∂ζ ∂U ∂V + + =0 ∂x ∂y ∂t (3) where u = U/H, v = V/H, H = h + ζ , ∂ ∂u ∂ ∂v ∂u Dx = AH 2 H + H + , ∂x ∂y ∂x ∂y ∂x ∂ ∂v ∂u ∂ ∂v Dy = AH H + + 2 H , ∂y ∂x ∂x ∂y ∂y where U and V are the volume transports in x- (alongchannel) and y- (cross-channel) directions, ζ is the sea surface elevation, h is the water depth, Dx and D y are the horizontal diffusions in x- and y-directions, κ is a friction coefficient (=0.0025), A H is horizontal eddy diffusivity (=100 m2s–1), f is the Coriolis parameter (9.5 × 10–5 s–1), and g is the gravity acceleration. Equations (1), (2), and (3) are solved numerically using a finite element method (Wang, 1975) with a horizontal resolution of 1 km. At the land boundaries the normal velocity is set to zero and a slip boundary condition is used. At the open boundaries both sea levels and currents are specified using a frictionally-damped progressive Kelvin wave solution (Park and Wang, 1991, 1994). The model is driven by a tidal wave propagating from left to right in the positive x-direction with an amplitude of 1 m and the maximum current speed of about 60 cm s–1. 3. Transient Vorticity Dynamics Cross-differentiating Eqs. (1) and (2), we obtain the transport vorticity equation; UU ∂ζ − κ ∇ × 2 ⋅ k + [ ∇ × D ] ⋅ k ∂t H IV V VI III ( 4) where Π (=∂ V/ ∂ x – ∂U/ ∂y) is the transport vorticity, N is the nonlinear advective term, J is the Jacobian operator, and k is the vertical unit vector. In Eq. (4), term I is the local acceleration of transport vorticity, term II is the vorticity advection, term III is the topographic vorticity tendency, term IV is the vorticity from sea surface divergence, term V is the bottom friction, and term VI is the vorticity diffusion. This equation may be compared to the vorticity of the depth-averaged velocity; uu f dH D ∂ω − κ ∇ × + ∇ ⋅ (uω ) = ⋅ k + ∇ × H ⋅ k H dt H ∂t A B C D E ω= Π 1 − (∇H × U ) ⋅ k H H2 (5) where ω (= ∂v/∂ x – ∂ u/∂ y) is the vorticity of the depthaveraged velocity. In Eq. (5) the term A is the local acceleration of the vorticity of the depth-averaged velocity, term B is the vorticity advection, term C is the water column stretching, term D is the bottom friction and term E is the vorticity diffusion. For low Rossby number flow with negligible bottom and lateral friction, the vortex stretching associated with the planetary vorticity, term C, will be the dominant vorticity production term. The (6 ) It consists of the depth-average of the transport vorticity and the correction term. The latter is comparable to the former for the flow parallel to the isobaths (Park and Wang, 1991). However, for the flow crossing the isobaths, it becomes small and the vorticity of the depth-averaged velocity may be approximated by the depth-average of the transport vorticity as discussed in Section 5. Comparing Eq. (4) to Eq. (5), it is evident that the terms I, II, V and VI of Eq. (4) correspond to terms A, B, D and E of Eq. (5). That is, the local acceleration, vorticity advection, bottom friction and vorticity diffusion have counterparts in both equations. Under the geostrophic approximation, the terms III and IV in Eq. (4) become − gJ ( H , ζ ) + f ∂Π + [∇ × N ] ⋅ k ∂t I II = − gJ ( H , ζ ) + f vorticity of the depth-averaged velocity is related to the transport vorticity; ∂ζ dH ≈ f ∂t dt which corresponds to term C of Eq. (5). This suggests that the topographic vorticity tendency may play the role of the column stretching when the transport crosses the isobaths producing the vorticity. The topographic vorticity tendency comes from the pressure gradient which is the dominant term in the equation for the tidal flow and its effect is shown explicitly in the transport vorticity equation, whereas in the vorticity equation of the depth-averaged flow it is implicit and appears through the continuity equation. Expansion of the friction term V of Eq. (4) yields, UU −κ ∇ × 2 ⋅ k H U U κ = −κ 2 Π + 2 [U × ∇ U ] ⋅ k − κ 4 U × ∇ H 2 ⋅ k H H H V −1 V−2 V−3 [ ( )] (7) The first term on the right hand side of Eq. (7) is the vorticity dissipation by bottom friction. The second and third terms are the frictional torque. Thus, Eq. (4) states that, following a fluid column the transport vorticity is produced by the topographic vorticity tendency (III), transport shear (V-2) and depth gradient (V-3), and is dissipated by the bottom friction (V-1) and horizontal diffusion (VI). The major forcing for the transient transport vorticity is the topographic vorticity tendency, which is induced when the isolines of the free surface cross the Tidal Vorticity Around a Coastal Promontory 263 Fig. 2. The instantaneous transport. (a) to (f) are 1/6 tidal cycle interval starting at the time of the maximum ebb. The maximum transport in (a) is 16.5 m2s –1. isobaths. The depth-integrated vorticity equation has been used for the large scale flows over a sloping bottom and the bottom pressure torque which corresponds to the topographic vorticity tendency in a homogeneous flow has been shown to be a dominant term in the transport vorticity equation (e.g. Holland, 1973; Ezer and Mellor, 1994). This effect is not isolated in the vorticity equation for the depthaveraged velocity (Park and Wang, 1991) and we will proceed hereafter with the transport vorticity equation. The instantaneous transport is shown at 1/6 tidal cycle interval in Fig. 2. The transport pattern at the time of the maximum ebb (Fig. 2(a)) is similar to that at the maximum flood (Fig. 2(d)) except the change of the flow 264 M.-J. Park and D.-P. Wang direction. The transport increases as flows approaching the tip of the promontory and decreases after flows passing the promontory. The increase of transport is mainly associated with the increase of currents around the promontory. For a fluid particle moving on a curved path, the pressure gradient across the fluid particle path is balanced by the centrifugal and Coriolis accelerations. The relative importance of the centrifugal acceleration with respect to the Coriolis acceleration is measured by the Rossby number (R o = S/fR). At the tip of the promonory, S ~ 90 cm s –1 and R ~ 4 km, Ro ~ 2.3. Thus, the centrifugal acceleration is more important than the Coriolis acceleration. As a fluid particle approaches the promontory, Fig. 3. Transient transport vorticity. (a) to (f) are in 1/6 tidal cycle interval starting at the time of the maximum ebb current. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 15 × 10 –4 m s–1. the centrifugal acceleration is balanced by a sea level depression. The maximum sea level depression is at the minimum radius of curvature, i.e. off the tip of the promontory. This sea surface slope induces an acceleration when fluid particles move toward the promontory, and a deceleration when fluid particles move away from the promontory. Hence, the maximum velocity is near the tip of the promontory. The instantaneous transport vorticity (Fig. 3) shows a positive vorticity during the ebb and a negative vorticity during the flood. The maximum positive vorticity is slightly stronger than the maximum negative vorticity. Also, the maximum positive and negative vorticities are found slightly downstream of the promontory. The topographic vorticity tendency, vorticity advective term, bottom friction and vorticity diffusion are examined at three locations around the tip of the promontory (see Fig. 1 for the locations). The bottom friction is further divided into the dissipation by bottom friction (V-1) and frictional torque (V-2 and V-3). The transient vorticity balance is shown over a tidal cycle in Figs. 4 and 5. The transient vorticity is generated by the topographic vorticity tendency and the frictional torque. The former is large off the tip of the promontory where the sea surface slope is Tidal Vorticity Around a Coastal Promontory 265 Fig. 4. Transient transport vorticity balance at stations A (top), B (middle), and C (bottom) (see Fig. 1 for the locations). The solid lines are for the vorticity advective term, the chaindotted lines are for the topographic vorticity tendency, the dashed lines are for the friction, and the dotted lines are for the vorticity diffusion. Note that the values were multiplied by 10 8. normal to the bottom slope (Fig. 4). The topographic vorticity tendency is mostly positive during the ebb and negative during the flood. The transient vorticity is also generated by the frictional torque. The frictional torque due to the depth gradient is positive during the ebb and negative during the flood (Fig. 5). The depth gradient contribution is largest off the tip of the promontory, at the maximum ebb or flood. The frictional torque generated by the depth gradient, on the other hand, tends to be cancelled by the frictional torque generated by the transport shear and frictional dissipation (Fig. 5). Consequently, the net effect due to bottom friction is small and the vorticity balance is mainly between the topographic 266 M.-J. Park and D.-P. Wang Fig. 5. The transport vorticity generation by the frictional torque at stations A (top), B (middle), and C (bottom) (see Fig. 1 for the locations). The solid lines are for the friction (sum of terms V-1, V-2 and V-3 of Eq. (7)), the chain-dotted lines are for the vorticity dissipation by bottom friction (term V1), the dashed lines are for the vorticity generation by transport shear (term V-2), and the dotted lines are for the vorticity generation by depth gradient (term V-3). Note that the values were multiplied by 10 8. vorticity tendency and the vorticity advective term (Fig. 4). The vorticity advective term becomes large at the maximum flood and ebb. It is almost always positive on the left-hand side of the promontory, as there is an influx of positive vorticity during the ebb and an efflux of negative vorticity during the flood. Similarly, it is mostly negative on the right-hand side. 4. Residual Vorticity Dynamics The residual transport vorticity and the residual transport are obtained by averaging their respective instanta- Fig. 6. (a) Residual transport vorticity. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 2.5 × 10–4 m s –1. (b) Residual transport. The maximum value is 2.0 m2s –1. Fig. 7. Residual transport vorticity balance. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 7 × 10 –8 m s –2. (a) Residual vorticity advective term, (b) residual topographic vorticity tendency, (c) residual friction, and (d) residual vorticity diffusion. neous values over a tidal cycle. The residual vorticity is positive (cyclonic) on the left-hand side of the promontory and negative (anticyclonic) on the right-hand side (Fig. 6(a)). The residual transport pattern shows a cyclonic eddy on the left-hand side of the promontory and an anticyclonic eddy on the right-hand side (Fig. 6(b)). Between the counter-rotating eddy pair, there is a strong offshore jet off the tip of the promontory. Because of the larger positive vorticity, the residual transport is stronger in the cyclonic eddy than in the anticyclonic eddy. In order to understand how the residual transport vorticity arises, we analyzed the residual vorticity balance. Averaging Eq. (4) over a tidal cycle, we obtained the residual transport vorticity equation. This consists of Tidal Vorticity Around a Coastal Promontory 267 the residual vorticity advective term, residual topographic vorticity tendency, residual friction, and residual vorticity diffusion; [∇ × N ] ⋅ k + gJ ( H , ζ ) UU = − κ ∇ × 2 H ⋅ k + [∇ × D ] ⋅ k (8) at the right-hand side (Fig. 7(a)). On the other hand, the residual topographic vorticity tendency is positive at the right-hand side and negative at the left-hand side (Fig. 7(b)), which tends to counterbalance the residual vorticity advective term. In other words, these are two dominant terms in the residual vorticity balance. The residual friction (Fig. 7(c)) and residual horizontal diffusion (Fig. 7(d)) are relatively small. 5. where 〈 〉 represents the time average over a tidal cycle. The dominant advective term is 〈u·∇Π〉 which, due to its nonlinearity, has a large residual when averaged over a tidal cycle. As noted before, the vorticity advective term during a tidal cycle is mostly positive at the left-hand side of the promontory and negative at the right-hand side (Fig. 4). Thus, the residual vorticity advection is positive (cyclonic) at the left-hand side and negative (anticyclonic) Comparison of Model Results with the Observations The model results can be compared with the observations at Gay Head (Geyer and Signell, 1990, 1991). Figure 8 shows the instantaneous vorticity ( ω = ∂v/ ∂x – ∂ u/∂ y) and current at the maximum flood from observations and model results. The calculated vorticity pattern is quite similar to the observation. Both show a negative vorticity off the tip of the promontory having the maxi- Fig. 8. Comparison of the observed instantaneous vorticity of depth-averaged velocity, and the observed currents (Geyer and Signell, 1991) with those from the model at the time of the maximum flood. The contour interval of the vorticity is 0.5 × 10 –4 s–1. The maximum velocity from the model is 0.87 m s–1. 268 M.-J. Park and D.-P. Wang mated by the depth-average of the transport vorticity. The residual currents from the observations show a cyclonic eddy at the left-hand side of the promontory and an anticyclonic eddy on the right-hand side (Fig. 9). The offshore extent of the jet is about 5 km, and the maximum residual current is about 25 cm s–1. The model results reveal a comparable pattern. However, the observations indicate a symmetric eddy pair, whereas the model shows an asymmetric pair with stronger cyclonic eddy. In the model the axis of the promontory is normal to the tidal stream, and hence the centrifugal force acting on flood currents is almost equal to that on ebb currents. Due to the opposing effect of Coriolis force on flood and ebb currents, however, the cyclonic eddy becomes stronger. On the other hand, the axis of Gay Head is inclined cyclonically towards the tidal stream axis. This provides a sharper curvature and a stronger centrifugal force for the flood current (Geyer, 1993). Perhaps, the oblique orientation of the headland axis to the tidal flow leads to a more symmetric residual eddies. Fig. 9. Comparison of the observed residual currents with the residual currents from the model. The maximum velocities are 25 cm s–1 from the observed data (Geyer and Signell, 1991) and 21 cm s –1 from the model. 6. Sediment Dynamics The tidal vorticities may play an important role in the sediment transport and sand bank formation. Previous studies on sediment transport around a coastal promontory were rather qualitative (Pingree, 1978; Pingree and Maddock, 1979), and were confined to the study of offshore sand bank formation (Pingree, 1978; Pingree and Maddock, 1979; Heathershaw and Hammond, 1980). We used a more quantitative approach to examine effects of the tidal vorticities on the sand transport around a coastal promontory and on the stability of the promontory itself. Following Huthnance (1982a, 1982b) and Park and Wang (1991, 1994), we assumed that the bed load transport rate Qb (volume/unit width/time) is related to |u|3 , Qb ~ u (u + λ u ∇h) 2 mum vorticity located at slightly downstream off the promontory. The model and observed vorticity dimensions also are comparable, about 8 km alongshore. The observed vorticity (–4.5 × 10–4 s –1), however, is about two times the model result (–2.4 × 10–4 s–1). By comparison, Signell and Geyer (1990, 1991) obtained the maximum vorticity for the flood current of –25 × 10–4 s–1 from their model. Also, both observed and modelled currents for the maximum flood show clearly the crossing of the isobaths where the maximum vorticity is located (Fig. 8). It is worthwhile to note that the pattern of the vorticity of the depthaveraged velocity at the maximum flood is similar to that of the transport vorticity (Fig. 2(d)). This results from the fact that the correction term on the right-hand side of Eq. (6) is smaller than the depth-average of the transport vorticity for the flow crossing the isobaths. Thus, the vorticity of the depth-averaged velocity may be approxi- ( 9) where the second term on the right-hand side of Eq. (9) is the enhancement of the down-slope transport and λ of 0.005 is used as in Huthnance (1982a, 1982b). We first examined the transient sediment transport process by computing the instantaneous divergence of bed load (∇·Qb ). The erosion is expected in areas of bed load divergence and the deposition in areas of bed load convergence. During the maximum ebb current, the velocity increases over the broad area of the slope toward the tip of the promontory, and the erosion occurs over the slope along the right-hand side of the promontory (Fig. 10(a)). As the ebb current passes over the promontory, however, the velocity decreases abruptly, and the deposition is concentrated behind the tip of the promontory resulting in an asymmetric pattern of erosion and deposition (Fig. 10(a)). During the maximum flood current, on the other hand, Tidal Vorticity Around a Coastal Promontory 269 Fig. 10. Instantaneous bed load divergence. (a) to (f) are in 1/6 tidal cycle interval starting at the time of the maximum ebb current. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 3 × 10–5 m 2s–3. the velocity changes across the tip of the promontory are more gradual. Thus, the erosion and deposition occur over a broad area, and the sedimentation pattern tends to be much more symmetric (Fig. 10(d)). The different patterns of erosion and deposition between the maximum flood and ebb currents are a manifestation of the asymmetric tidal currents. The centrifugal acceleration is partly canceled by the Coriolis acceleration during the flood, but it is reinforced during the ebb. This creates a stronger positive vorticty during the ebb currents, which results in a more concentrated deposition near the tip of the promontory. 270 M.-J. Park and D.-P. Wang The residual bed load divergence was obtained from averaging over a tidal cycle the instantaneous values. It has a net erosion along the left-hand side (upstream) slope of the promontory and a net deposition on the slope off the tip of the promontory (Fig. 11(a)). The deposition also occurs outside the slope on areas where the residual eddies are formed (Fig. 11(b)), but the deposition rate is an order of magnitude smaller than that off the tip of the promontory. Previous studies (e.g. Pingree and Maddock, 1979) discussed only the relationship between the residual eddies and the offshore bank formation around a promontory. The result of this study shows that the net sedi- Fig. 11. Residual bed load divergence. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. (a) The contour interval is 10 × 10 –6 m2s –3. (b) The contour interval is 1 × 10–6 m 2s–3. ment transport associated with the coastal promontory itself is an order of magnitude larger than that for the offshore bank formation. This net erosion and deposition pattern suggests that the promontory will tend to incline cyclonically from the axis normal to the tidal stream. Indeed, as discussed earlier, Gay Head is such an example. The cyclonic inclination of a promontory is also found at Portland Bill in English Channel. Also, for an asymmetric promontory inclining cyclonically from the axis normal to the tidal stream, the centrifugal acceleration is enhanced for the flood current and reduced for the ebb current. This may result in more symmetric residual eddies, as found in Portland Bill (Pingree, 1978; Pingree and Maddock, 1979) and Gay Head (Geyer and Signell, 1990). In other words, an asymmetric promontory may be a more stable configuration (Park, 1998) with a symmetric eddy pair. On the other hand, a symmetric promontory with the axis orientation normal to the tidal stream may not be stable until it deforms into a shape by which the resulting residual eddies become symmetric. 7. Discussion The transport vorticity equation was used to study the topographic vorticity generation by the tidal currents around a coastal promontory in shallow coastal seas. Our results indicate the presence of strong transient vorticities off the tip of a promontory. The vorticity analysis shows that the transient vorticities are generated mainly by the topographic vorticity tendency, which is consistent with the results on other isolated bottom topographic features such as hollow and sand banks (Park and Wang, 1991, 1994). In contrast, previous model studies suggested forcing by lateral friction or frictional torque. Tee (1976), for example, indicated that a no-slip boundary condition is essential for the vorticity generation around a promontory. Yanagi (1976, 1978) and Oonishi (1977) also emphasized the lateral friction. Signell and Geyer (1991) adopted similar approach, but they used higher resolution. Our study, however does not depend on a “wall” boundary. Since we usually find in nature a bottom sloping upward to the coast in a coastal promontory as in our study, this mechanism may not be different from the bottom frictional torque (Zimmerman, 1981). Pingree and Maddock (1979), Ridderinkhof (1989) and Ridderinkhof and Zimmerman (1990), on the other hand, suggested that the frictional torque is important for the transient vorticity generation around a coastal promontory. We found that the frictional torque associated with the depth gradient is indeed large. However, it is counterbalanced by the frictional torque associated with the transport shear and the net effect is smaller than the topographic vorticity tendency. In fact, it was also noted that the vorticity generation by the depth gradient is opposed by the velocity shear (Pingree and Maddock, 1979), and the combined effect of the vorticity generation by the depth gradient and the velocity shear in the vorticity equation of the depth-averaged flow is small (Ridderinkhof, 1989; Ridderinkhof and Zimmerman, 1990). On the other hand, the magnitude of the column stretching associated with the relative vorticity (ω ∇·u) was found to be comparable to the combined effect of the frictional torque (Ridderinkhof, 1989). This column stretching mechanism may play a role analogous to the topographic vorticity tendency for the vorticity generation around a coastal promontory where the advective acceleration is more important than the Coriolis acceleration (Ro ~ 2.3 in our study). The residual vorticity balance of the vorticity of the depth-averaged flow also shows that the vorticity advection is largely balanced by the column stretching (Ridderinkhof, 1989) which is analogous to the residual transport vorticty balance. These results appear to be consistent with the fact that the vorticity of the depth-averaged velocity in this study may be approximated by the depth-average of the transport vorticity as described in Section 5. Tidal Vorticity Around a Coastal Promontory 271 The present study investigates a coastal promontory that is quite typical in shallow coastal seas. The length scale of the promontory is comparable to the tidal excursion amplitude, that is, the inertia ratio is O(1). This is an essential condition for the maximum vorticity generation (e.g. Zimmerman, 1981; Robinson, 1981). The Reynolds number (R e = H/ κw) is about 4.2 and the inertia ratio (K = U/ σw) is about 2.1 in our study, where σ is the tidal frequency and w is the width scale of the promontory. Since the Reynolds number is comparable to the inertia ratio, the frictional decay length (L = H/2κ ~ 4.2 km) is comparable to the tidal excursion (E = 2U/σ ~ 8.5 km). In other words, while the time-dependent effects are important, friction is strong enough so that the vorticity decays over the course of a tidal cycle (Signell and Geyer, 1991). Increasing the water depth increases the Reynolds number, that is, the advective acceleration will dominate the friction. Signell and Geyer (1991) have explored the flow dynamics in the high Reynolds number limit. Although the dynamics of flow separation is quite interesting, it is clearly beyond the scope of the present work. Increasing tidal velocity or promonotory curvature (i.e. to increase the Rossby number) will make the advective acceleration much more important than the Coriolis acceleration, and the residual eddies may become more symmetric. The three-dimensional effect on the sand transport is ignored in this study. On the other hand, the secondary circulation with offshore flow near the surface and compensating return flow near the bottom is common around the tip of a headland (Heathershaw and Hammond, 1980; Geyer, 1993). Deleersnijder et al. (1992) and Wolanski et al. (1996) used three-dimensional models to investigate the flow structure around a small island. They suggested that upwelling may occur in the center of eddies. However, the magnitude of the secondary circulation is an order of magnitude smaller than the primary current (Heathershaw and Hammond, 1980; Deleersnijder et al., 1992; Geyer, 1993). This suggests that the secondary circulation effect may be of minor importance in this study of sand transport. Nevertheless, to fully explore the threedimensional effect of the flow on the sediment transport around a coastal promontory, we may need a three-dimensional model. Acknowledgements This study was partially supported by the grants (1998-022-H00009 and 1997-013-147) from Korean Ministry of Education. M.-J. P. would like to appreciate the hospitality of the personnel of Marine Sciences Research Center, State University of New York, Stony Brook during his visit as a visiting professor in 1997–1998 academic year where a part of this study was done. We also appreciate useful comments from reviewers. 272 M.-J. Park and D.-P. 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